Next month’s Notices of the AMS has an essay by Freeman Dyson entitled Frogs and Birds, which was written for his planned Einstein Public Lecture. In it, he divides mathematicians up into two species: birds, who “fly high in the air and survey broad vistas” (i.e. seek abstraction, unification and generalization), and frogs, who “see only the flowers that grow nearby” (i.e. study the details of specific examples).
Dyson himself is resolutely a frog, but writes that “many of my best friends are birds”, and argues that both birds and frogs are needed to do justice to the breadth and depth of the subject of mathematics. Frog that he is, his essay covers a variety of quite different special topics that have drawn his attention, linked together only weakly by the bird/frog theme. These include a discussion of the roles of complex numbers and linearity in quantum mechanics, a proposed idea about how to attack the Riemann hypothesis (try and enumerate 1d-quasicrystals, since the zeros of the zeta function have this structure), and a collection of profiles and anecdotes about various mathematicians and physicists (Besicovitch, Weyl, Yang, Manin, von Neumann).
Personally I suppose I fit well into Dyson’s bird category, but among the best mathematicians that I know, the frog/bird distinction is often unclear. Many of them make their reputation by proving rather abstract and general theorems, but these proofs are often the result of a huge amount of detailed investigation of examples. I agree with Dyson that both points of view are needed, and see the most successful cases of progress in mathematics coming from mathematicians who avoid the temptation to fly too high into arid abstraction, or sink too deep into irrelevant detail.
Dyson includes a long section on string theory, which I’ll include here:
I would like to say a few words about string theory. Few words, because I know very little about string theory. I never took the trouble to learn the subject or to work on it myself. But when I am at home at the Institute for Advanced Study in Princeton, I am surrounded by string theorists, and I sometimes listen to their conversations. Occasionally I understand a little of what they are saying. Three things are clear. First, what they are doing is first-rate mathematics. The leading pure mathematicians, people like Michael Atiyah and Isadore Singer, love it. It has opened up a whole new branch of mathematics, with new ideas and new problems. Most remarkably, it gave the mathematicians new methods to solve old problems that were previously unsolvable. Second, the string theorists think of themselves as physicists rather than mathematicians. They believe that their theory describes something real in the physical world. And third, there is not yet any proof that the theory is relevant to physics. The theory is not yet testable by experiment. The theory remains in a world of its own, detached from the rest of physics. String theorists make strenuous efforts to deduce consequences of the theory that might be testable in the real world, so far without success.
My colleagues Ed Witten and Juan Maldacena and others who created string theory are birds, flying high and seeing grand visions of distant ranges of mountains. The thousands of humbler practitioners of string theory in universities around the world are frogs, exploring fine details of the mathematical structures that birds first saw on the horizon. My anxieties about string theory are sociological rather than scientific. It is a glorious thing to be one of the first thousand string theorists, discovering new connections and pioneering new methods. It is not so glorious to be one of the second thousand or one of the tenth thousand. There are now about ten thousand string theorists scattered around the world. This is a dangerous situation for the tenth thousand and perhaps also for the second thousand. It may happen unpredictably that the fashion changes and string theory becomes unfashionable. Then it could happen that nine thousand string theorists lose their jobs. They have been trained in a narrow specialty, and they may be unemployable in other fields of science.
Why are so many young people attracted to string theory? The attraction is partly intellectual. String theory is daring and mathematically elegant. But the attraction is also sociological. String theory is attractive because it offers jobs. And why are so many jobs offered in string theory? Because string theory is cheap. If you are the chairperson of a physics department in a remote place without much money, you cannot afford to build a modern laboratory to do experimental physics, but you can afford to hire a couple of string theorists. So you offer a couple of jobs in string theory, and you have a modern physics department. The temptations are strong for the chairperson to offer such jobs and for the young people to accept them. This is a hazardous situation for the young people and also for the future of science. I am not saying that we should discourage young people from working in string theory if they find it exciting. I am saying that we should offer them alternatives, so that they are not pushed into string theory by economic necessity.
Finally, I give you my own guess for the future of string theory. My guess is probably wrong. I have no illusion that I can predict the future. I tell you my guess, just to give you something to think about. I consider it unlikely that string theory will turn out to be either totally successful or totally useless. By totally successful I mean that it is a complete theory of physics, explaining all the details of particles and their interactions. By totally useless I mean that it remains a beautiful piece of pure mathematics. My guess is that string theory will end somewhere between complete success and failure. I guess that it will be like the theory of Lie groups, which Sophus Lie created in the nineteenth century as a mathematical framework for classical physics. So long as physics remained classical, Lie groups remained a failure. They were a solution looking for a problem. But then, fifty years later, the quantum revolution transformed physics, and Lie algebras found their proper place. They became the key to understanding the central role of symmetries in the quantum world. I expect that fifty or a hundred years from now another revolution in physics will happen, introducing new concepts of which we now have no inkling, and the new concepts will give string theory a new meaning. After that, string theory will suddenly find its proper place in the universe, making testable statements about the real world. I warn you that this guess about the future is probably wrong. It has the virtue of being falsifiable, which according to Karl Popper is the hallmark of a scientific statement. It may be demolished tomorrow by some discovery coming out of the Large Hadron Collider in Geneva.
I don’t know where Dyson got the estimate of ten thousand string theorists; my own estimate would be more like one to two thousand (with the number strongly dependent on how you decide who is a “string theorist”). The large yearly Strings200X conferences that bring together a sizable fraction of active string theory community tend to draw roughly 500 people.
The Princeton-centric assumption that there are lots of string theory jobs embedded in his question “And why are so many jobs offered in string theory?” is quite problematic, as any young string theorist on the job market could explain to him. There actually aren’t a lot of string theory jobs out there, and a lot of Ph.D.s in the subject being produced, leading to a lot of ex-string theorists now working in the financial industry and elsewhere. These days, if you are going to choose your field based on where the jobs are, you become an LHC phenomenologist or a cosmologist. If you want to be a string theorist, you better be a string phenomenologist or a string cosmologist. Also rather unrealistic is Dyson’s “it could happen that nine thousand string theorists lose their jobs”, due to tenure in the academic system. Even if a consensus develops over the next few years that string theory was all a big mistake, twenty years from now there will still be a cadre of (older) people working in the field.
Dyson’s idea, that 50-100 years from now, a new revolution in physics will show how string theory fits in may be right. It also may be that this has already happened, as much of the field has moved into the study of gauge-string dualities, where string theory provides a useful approximation for strongly coupled systems, and the idea that it unifies particle physics is falling by the wayside.